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DISCOVERY OF Φ AND π IN GIZA GREAT PYRAMID
By Prof. L. Kaliambos (Natural Philosopher in New Energy) January 8, 2016 A careful analysis of the dimensions of Giza great pyramid led to my formulation of π = 12/3Φ0.5 - 0.003 used for the construction of the great pyramid by relating the mathematical constant π = C/d to the golden section Φ = (1 + 50.5)/2 = (α+β)/α = α/β. Here the difference 12/3Φ0.5 - C/d = 0.003 could be found under a detailed measurement of the circumference C of the inscribed circle in the square of the square pyramid before the construction of it. Surprisingly, this formula includes the four mystic numbers π, Φ, 12, and 3 used by Dinocrates for the construction of the mathematical cone pyramid in Amphipolis for the divine hero Hephaestion. It means that in Spring of 323 BC Alexander the Great received from the oracle of Amon such mystic numbers. In other words the mathematical tomb of hero Hephaestion (320BC) is the miniature not only of Alexandria (331 BC) but also of the Great Pyramid at Giza (2560 BC). ''' Golden Section Φ is obtained by dividing a line into two parts (α and β) such that the square of the first part is equal to the product of the whole segment (α+β) and the second part. That is α2 = (α +β )β or (α +β )/α = α/β = Φ One method for finding the value of Φ is to start with the left fraction. Through simplifying the fraction and substituting in β/α = 1/Φ, (α+β)/α = 1 + β/α = 1+ 1/Φ Therefore 1 + 1/Φ = Φ and multiplying by Φ gives Φ + 1 = Φ2 which can be rearranged to Φ2 - Φ -1 = 0. Using this quadratic formula the solution is obtained as Φ = (1 + 50.5)/2 = 1.6180339887…. The Greeks, who called Φ the Golden Section, based the entire design of the Parthenon on this proportion. Phidias (500 BC - 432 BC), a Greek sculptor and mathematician, studied Φ and applied it to the design of sculptures for the Parthenon. Plato (circa 428 BC - 347 BC), in his views on natural science and cosmology presented in his "Timaeus," considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos. Moreover Dinocrates for the two Caryatids in the Hephaestion tomb in Amphipolis used the same golden section. ( See my “DINOCRATES”). The Egyptians used (Φ) in the design of the Great Pyramids and they thought that the golden ratio was sacred. Therefore, they used the golden ratio when building temples and places for the dead. The Egyptians were aware that they were using the golden ratio Φ, but they called it the "sacred ratio." There is debate as to the geometry used in the design of the Great Pyramid of Giza in Egypt. Built around 2560 BC, its once flat, smooth outer shell is gone and all that remains is the roughly-shaped inner core, so it is difficult to know with absolute certainty. The outer shell remains though at the cone, so this does help to establish the original dimensions. According to Wikipedia, the Great Pyramid as a square pyramid has a base a square with a side length α = 230.34 meters and an estimated original height h = 146.5 meters. If indeed exactly α = 230.34 meters then a perfect golden ratio would have a height of 146.5 m. In this case when a circle of a circumference C = 2πr inscribed in this square the diameter d = 2r is equal to the side length α = 230.34 m. Then the slant height L of the inscribed cone pyramid is found by using the Pythagorean theorem as L2 = h2 + r2 . Then writing L= Φr, and h = Φ0.5r one gets Φ2r2 = Φr2 + r2 or Φ2 = Φ +1 or Φ2 - Φ -1 = 0 in which Φ = (1 + 50.5)/2 = 1.618034 or Φ0.5 = 1.2720196 Since h = Φ0.5r and r = α/2 we get h = (1.272...)(230.34/2) = 146.5 m In addition to this relationship of the pyramid’s geometry to Φ it’s also possible that the pyramid was constructed by using the ratio C/d = C/α = π of the inscribed circle in the square pyramid. Before the construction of the pyramid they determined the side length (α) of the square pyramid on the ground and the inscribed circle of a circumference C = πd = πα. Thus measuring carefully the C and the α they were able to approximate the value of π close to π = 3.1415927. It was John Taylor (1859) who first proposed the idea that the number π = 3.1415927 might have been intentionally incorporated into the design of the Great Pyramid of Khufu at Giza. He discovered that if one divides the half perimeter (P/2) of the Pyramid by its height one obtains a close approximation to π. Indeed using the dimensions of the great pyramid and the value of Φ0.5 I discovered that the ratio P/2 to h cannot give the true value of π. In this case dividing P/2 = 2α = 460.72 m by its height h = 146.5 m, one obtains 2α/h = 460.72/146.5 = 3.1446... which is a close approximation to π = 3.1415927. Then it is possible to formulate a formula relating the constant π = C/d = C/α to Φ = (1+50.5)/2 Since h = Φ0.5( α/2) we may write 2α/h = 12/3Φ0.5 = 3.1446.. > π = 3.1415927 Nevertheless this formula is very useful because it can give us the difference 12/3Φ0.5 - C/α =''' '''3.1446.. -3.1415927 = 0.003 ' Under this condition in such a construction of the pyramid we can formulate the following relationship 'π = 12/3Φ0.5 - 0.003 ''' Surprisingly this formula includes the four mystic numbers π, Φ, 12, and 3 used by Dinocrates in Ampipolis. (See my "PLAN OF AMPHIPOLIS TOMB"). It is of interest to note that Egyptians at the time of 2560 BC for computing the constant π for the construction of the pyramid should be able to use such a method. Ancient civilizations needed the value of π to be computed accurately for practical reasons. However in Rhind Papyrus dated around 1650 BC we see a formula for the area of the circle that treats π as (16/9)2 = 3.1605 Of course my discovery of the relationship between π and Φ connects the dimensions of the Great pyramid at Giza. Much more later Archimedes (250 BC) using polygons computed in general upper and lower bounds of π and by calculating the perimeters of these polygons, he proved that 223/71 <π < 22/7. Another simple infinite series of π is the Gregory-Leibniz series: π = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 …. Category:Fundamental physics concepts